On the spectrum of stochastic perturbations of the shift and Julia sets
type: Item
instance of: scholarly article
title: On the spectrum of stochastic perturbations of the shift and Julia sets
zbMATH Open document ID: 1269.47004
author: Ali Messaoudi
author: El Houcein El Abdalaoui
published in: Fundamenta Mathematicae
publication date: 2012-07-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.6899
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 37A10
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 6055440
zbMATH Keywords: Markov operator
zbMATH Keywords: Markov process
zbMATH Keywords: stochastic adding machine
zbMATH Keywords: Julia sets
zbMATH Keywords: residual spectrum
zbMATH Keywords: continuous spectrum
MaRDI profile type: Publication
OpenAlex ID: W2023531498
Wikidata QID: Q120784072
arXiv ID: 1109.6899
DOI: 10.4064/FM218-1-3
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review text: The authors study the spectrum of some stochastic perturbation of the shift operator introduced by \textit{P. R. Killeen} and \textit{T. J. Taylor} [Nonlinearity 13, No. 6, 1889--1903 (2000; Zbl 0978.37001)], who considered a stochastic adding machine as a stochastic perturbation of the shift.NEWLINENEWLINEThe main results of the present paper are stated in the following three theorems.NEWLINENEWLINETheorem 3.1. The spectrum of the operator \(S_p\) acting on \(c_0, c\), and \(\ell^{\alpha }\), \(\alpha \geq 1\), is equal to the filled Julia set \(K_f\) of some quadratic map.NEWLINENEWLINETheorem 3.2. In \(\ell^1\), the point spectrum of \(S_p\) is empty. The residual spectrum of \(S_p\) is not empty and contains a countable dense subset of the Julia set \(J_f = \partial K_f\). The continuous spectrum of \(S_p\) is the complement of the residual spectrum in the Julia set \(K_f\).NEWLINENEWLINETheorem 3.3. The spectra of \(S_p\) acting, respectively, in \(\ell^{\infty }, c_0, c\) and \(\ell^{\alpha }\), \({\alpha } \geq 1\), associated to stochastic Fibonacci adding machines, contain a certain set (and are conjectured to equal this set).
reviewed by: Victor Sharapov
description: scientific article; zbMATH DE number 6055440